Positivity preserving along a flow over projective bundle

TL;DR

This paper introduces a flow on projective bundles that generalizes known flows and proves the preservation of curvature semipositivity under certain conditions, with applications to Hermitian-Yang-Mills and K"ahler-Ricci flows.

Contribution

It defines a new flow over projective bundles and proves curvature semipositivity preservation, extending results for Hermitian-Yang-Mills and K"ahler-Ricci flows.

Findings

Semipositivity of curvature is preserved along the flow under null eigenvector assumption.

Semipositivity is preserved when the base manifold is a curve.

Nonnegativity of holomorphic bisectional curvature is preserved under K"ahler-Ricci flow.

Abstract

In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and K\"ahler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under K\"ahler-Ricci flow.